Even though defined as formal power series, it is often useful to consider that generating functions are analytic functions of the complex variable \(z\) , in a small disk of convergence around the origin of the complex plane. This sometimes allows to find a closed form for the generating function in its disk of convergence, but not always. Even in this least favorable case, it enables to inherit fundamental results from complex analysis, which can be used for the purpose of enumerating combinatorial objects. In particular, we have in this tool box the Singular Inversion Theorem (Theorem VI.6 of [1]}),
which allows to derive asymptotic estimates of the coefficients of generating functions.
